A006428 Number of loopless tree-rooted planar maps with 3 vertices and n faces and no isthmuses.

0, 3, 36, 135, 360, 798, 1568, 2826, 4770, 7645, 11748, 17433, 25116, 35280, 48480, 65348, 86598, 113031, 145540, 185115, 232848, 289938, 357696, 437550, 531050, 639873, 765828, 910861, 1077060, 1266660, 1482048, 1725768, 2000526, 2309195, 2654820, 3040623

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Column 3 of A342985.

Programs

Mathematica

A006428[n_] := If[n == 1, 0, n*(n + 1)*(n*(n*(n + 6) + 11) - 42)/24];
Array[A006428, 50] (* Paolo Xausa, Aug 20 2025, after Andrew Howroyd *)

PARI

a(n) = if(n<2, 0, n*(n+1)*(n^3+6*n^2+11*n-42) / 24) \\ Andrew Howroyd, Apr 03 2021

Formula

a(n) seems to be divisible by n+1. - Ralf Stephan, Sep 01 2003

Conjecture (for n > 1): a(n) = n*(n+1)*(n^3+6*n^2+11*n-42) / 24. - Sean A. Irvine, Apr 10 2017

The above conjectures are true. - Andrew Howroyd, Apr 03 2021

From Chai Wah Wu, Aug 08 2022: (Start)

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 7.

G.f.: x^2*(2*x^5 - 12*x^4 + 30*x^3 - 36*x^2 + 18*x + 3)/(x - 1)^6. (End)

Extensions

Title improved by Sean A. Irvine, Apr 10 2017

Terms a(13) and beyond from Andrew Howroyd, Apr 03 2021